I have three variables $x_1, x_2, x_3$ that can only be non-negative integers, i.e., $x_1 \in \mathbb{N}_0$, $x_2 \in \mathbb{N}_0$, and $x_3 \in \mathbb{N}_0$. Is it correct to write this in a tuple like $(x_1,x_2,x_3) \in \mathbb{N}_0^3$? If not, what is the correct way?
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$\begingroup$ that is correct $\endgroup$– J. W. TannerMar 30, 2023 at 3:15
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4$\begingroup$ Yes, or $(\Bbb N_0)^3$ for more clarity. $\endgroup$– azif00Mar 30, 2023 at 3:26
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2$\begingroup$ Another option for writing it is to enclose the $\mathbb{N}_0$ in braces, giving you ${\mathbb{N}_0}^3$. I'm not sure I actually like that, though. $\endgroup$– Tanner SwettMar 30, 2023 at 18:28
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1$\begingroup$ @TannerSwett - Oh, you mean add braces in the underlying LaTeX, so that the superscript isn't directly over the subscript! I kept thinking "I don't see any braces" and wondering if my rendering was broken. $\endgroup$– JonathanZMar 31, 2023 at 17:45
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1$\begingroup$ @JonathanZsupportsMonicaC Yup, sorry for my ambiguous writing! $\endgroup$– Tanner SwettMar 31, 2023 at 23:16
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2 Answers
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Unless you're actually considering them as a tuple elsewhere, you can also just write this as
$$x_1, x_2, x_3 \in \mathbb N_0$$
And I would argue that introducing the tuple and the product space solely for the purpose of specifying this constraint makes it more complicated for no reason.
So if you need the tuple, yes, use that notation. But if you don't need the tuple, use this one.
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Yes, this is correct. You could also write $$(x_1,x_2,x_3)\in\mathbb{N}_0 \times \mathbb{N}_0 \times \mathbb{N}_0.$$
An alternative that avoids duplication is $$\text{$x_i\in \mathbb{N}_0$ for $i\in \{1,2,3\}$}.$$
